High Precision Measurements of the Form Factors of Pion, Kaon, and Proton at Large Timelike Momentum Transfers

0.17 MB·English

by  Kamal K. Seth

Electromagnetic Structure of Proton, Pion, and Kaon by High Precision
Measurements of their Form Factors at Large Timelike Momentum Transfers
Kamal K. Seth,1 S. Dobbs,1 Z. Metreveli,1 A. Tomaradze,1 T. Xiao,1 and G. Bonvicini2
1Northwestern University, Evanston, Illinois 60208, USA
2Wayne State University, Detroit, Michigan 48202, USA
(Dated: December 20, 2012)
The electromagnetic structure of the lightest hadrons, proton, pion, and kaon, is studied by
high precision measurements of their form factors for the highest timelike momentum transfers of
Q2 = s = 14.2 and 17.4 GeV2. Data taken with the CLEO-c detector at √s = 3.772 GeV and
|4.17|0 GeV,with integrated luminosities of 805 pb−1 and 586 pb−1, respectively,havebeen used to
studye+e−annihilationsintoπ+π−,K+K−,andpp¯. Thedimensionalcountingrulepredictionthat
2 atlargeQ2 thequantityQ2F(Q2)forpseudoscalarmesonsisnearlyconstant,andshouldvaryonly
01 weaklyasthestrongcouplingconstant,αS(Q2),isconfirmedforbothpionsandkaons. However,the
measurements are in strong quantitative disagreement with the predictions of the existing QCD–
2
based models. For protons, it is found that the timelike form factors continue to remain nearly

c twiceaslargeasthecorrespondingspacelikeformfactors measuredinelectronelasticscattering, in
e significant violation of the expectation of their equality at large Q2. Further, in contrast to pions
D
and kaons, a significant difference is observed between the values of the corresponding quantity
19 |oQf4|Q|G2M|G(M|Q(2|Q|)/2|µ)p/µfopr,pinrsotteoands,aatt|Qth2e|s=e l1a4rg.2eG|QeV2|2. and17.4GeV2. Theresultssuggesttheconstancy

PACSnumbers: 13.40Gp,14.20Dh,14.40Be,14.40Df
]
x
e
- Knowledge of the quark–gluon structure of the only formfactorswerefactors4 8largerthanpredicted. Fur-
p −
stable baryon, the proton, and the lightest mesons, the ther, the ratio F (Q2 )/F (Q2 ) was also found to be
e π K
| | | |
h pion and kaon, is of great interest for both nuclear and nearlytwiceaslargeastheQCDpredictionthatitshould
[ particle physics. Important questions about the size of beequaltotheratioofthesquaresofthedecayconstants,

the proton, the composition of its spin, and the large f2/f2 at large Q2 [7]. These large differences from the-
2 π K
v difference between its spacelike and timelike form fac- oreticalpredictions raiseimportant questions about how
6 tors remain open. Timelike form factors of pions and validtheasymptoticpredictionsareatthemeasuredmo-
9 kaons, which are needed for the precision determination mentum transfers, and make it imperative to extend the
5
of the hadronic loop contribution to the muon g 2 measurements to larger momentum transfers.
1 −
anomaly [1, 2], are poorly known. Spacelike form fac- We use data taken with the CLEO-c detector, which
.
0
tors of pions and kaons needed for the understanding of has been described in detail before [8], to study the re-
1 nuclear and hypernuclear forces are difficult to measure actionse+e− pp¯, π+π−, andK+K−. The mainbody
2
1 at large momentum transfers, and can only be obtained of the data co→mprise of e+e− annihilations at center-of-
: by analytic continuation of timelike form factors [3]. To mass energies √s = 3772 MeV (Q2 = 14.2 GeV2) and
v | |
meet these needs, precision measurements of timelike 4170 MeV (Q2 = 17.4 GeV2), with integrated lumi-
i
X formfactorsatthe highestpossible momentum transfers nosities of | =|805 pb−1 and = 586 pb−1, respec-
L L
r are needed. In this Letter we report measurements of tively. Data from a mini–scan at the average √s =
a the form factors of pions, kaons, and protons with much 4010.4 MeV ( Q2 = 16.08 GeV2, = 20.7 pb−1) and
higherprecision,andformuchlargertimelikemomentum √s = 4260 Mhe|V (|iQ2 = 18.25 GeVL2, = 12.9 pb−1)
| | L
transfers than before [4, 5]. have also been analyzed.
Earlier measurements of proton form factors for large No measurements of the branching fractions for non–
timelike momentumtransfers(Q2 <0)made by the Fer- DD two–body decays of either ψ(3772) or ψ(4160) to
milab E760/E835 pp¯ e+e− experiments for Q2 = light hadrons exist [9], but they can be estimated us-
8.84 13.11 GeV2 [4→], and the CLEO e+e− | | pp¯ ing the perturbative QCD (pQCD) prediction that the
− →
measurements at Q2 = 13.48 GeV2 [5], revealed that hadronic decays of ψ(nS) states scale with the princi-
| |
the timelike form factors are nearly twice as large as ple quantum number n in the same way as the dilep-
the corresponding spacelike form factors, a result in ton decays. The measured values, (ψ(3770) e+e−)
strong disagreementwith the expectation of their equal- and (ψ(4160) e+e−), are 103Btimes sma→ller than
ity [6] at asymptotically large Q2 . The measurement (ψ(B2S) e+e→−) [9]. This le≈ads to the estimate that
of pion and kaon timelike form| f|actors by CLEO at tBhebranch→ingfractionsforπ+π−,K+K−,andpp¯decays
Q2 =13.48GeV2 [5]revealedthatwhilethe dimensional of ψ(3770) and ψ(4160) are approximately 0.9 10−8,
counting rule prediction of a α /Q2 variation of the 9 10−8,and3.2 10−7,respectively,basedont×he mea-
S
| | × ×
form factors [6] was apparently confirmed, the measured surements for ψ(2S) decays [9, 10]. This leads to the
2
estimates that with more than 5.2 million ψ(3772) and For particle identification in the RICH detector,
ψ(4160) formed in the present measurements, the num- we define the parameter L , which is the likelihood
i
bers of resonantly produced π+π−, K+K−, and pp¯ are that a particle corresponds with a given hypothesis
0.04, 0.4, and 1.8, respectively. In other words, the of being of species, i, based on the Cherenkov pho-
∼
resonance contribution to their yield is expected to be tons detected in the RICH detector. We combine
vanishinglysmall,andthecountsobservedinthepresent this with dE/dx information from the drift chambers,
analysis can be entirely attributed to the timelike elec- χ2(dE/dx), defined as χ2(dE/dx) = [(dE/dx)
measured
−
tromagnetic form factors. (dE/dx) ]2/σ2 , to construct the joint vari-
expected measured
The event selection and particle identification for the able ∆ (i,j) to distinguish particle type i from contam-
L
analysis reported in this Letter are similar to those in inant particle type j,
our earlier publication [5]. The reconstructed events are
requiredtohavetwochargedtracks,zeronetcharge,and
cosθ <0.80. Eachchargedparticleisrequiredtosatisfy ∆ (i,j)= 2[log(Li) log(Lj)]
| | L − −
the standard criteria for track quality and origin of the +[χ2(dE/dx) χ2(dE/dx) ]. (3)
i j
track at the interaction point. In order to develop data- −
independent particle identificationcriteria,andto deter-
mineeventselectionefficiencies,MonteCarlo(MC)sam-
Monte Carlo simulations show that the default values
ples weregeneratedfor e+e− h+h−, h=p,π,K using
of ∆ (i,j) < 0 are very effective in distinguishing be-
the EvtGen generator [11], an→d e+e− l+l−, l = e,µ, L
tweenthedesiredparticleiandthecontaminantparticle
using the Babayaga generator [12]. Fo→r e+e− π+π−
j. Accordingly,werequire∆ (p,µ),∆ (p,e),∆ (K,µ),
and K+K−, the Monte Carlo samples were g→enerated L L L
∆ (K,e), ∆ (K,π), ∆ (K,p), ∆ (π,e), ∆ (π,K),
with sin2θ angular distributions, since for pseudoscalar L L L L L
∆ (π,p) < 0. Because pions and muons have similar
mesons, the differential cross sections are L
masses, a ∆ (i,j) cut is not effective in distinguishing
L
between pions and muons. It was shown in Ref. [5] that
dσ (s,θ) /dΩ=(α2/8s)β3 F (s)2sin2θ, (1)
0 m m| m | ECC canbeveryeffectiveindistinguishingmuons,which
sufferonlyionizationlossinthe CC crystals,frompions,
where α is the fine structure constant, cβ is the meson
m
which suffer additional energy loss due to hadronic in-
velocity in the laboratory frame, and F (s) is the form
m
teractionsin the crystals. We thereforeimpose the addi-
factor for timelike momentum transfer at s = Q2 . For
e+e− pp¯theMonteCarlosamplesweregener|ated| with tional requirement of ECC > 350 MeV. With this addi-
both (→1+cos2θ) and sin2θ angular distributions, since tionalrequirement,muoncontaminationinthepionpeak
isfoundtobe lessthan1%,andthe X distributions
π,K,p
for both √s = 3772 MeV and 4170 MeV are all free of
dσ (s,θ) /dΩ=(α2/4s)β [G (s)2(1+cos2θ)
0 p p | M | contaminants, as shown in Fig. 1.
+τ G (s)2sin2θ], (2)
| E | Since the peaks in Fig. 1 have essentially no back-
grounds, we obtain the number of counts N (s) and
where G (s) and G (s) are the electric and magnetic π
E M
form factors of the proton, respectively, and τ ≡4m2p/s. aNnKd(sN) (ass)caosunctosunintstihnethraenrgaengXeπ,XK == 11..000000± 00..001150,.
Monte Carlo generated momentum distributions for p p ±
different pairs of hadrons (h+h−) and leptons (l+l−) We note that the Xp distribution for √s = 3772 MeV
in Fig. 1(e) has a definite tail in the low X region. As
show that the peaks for individual particles are broad- p
shownbytheunshadedhistograminFig.1(e),thisarises
enedbydetectorresolution,andtheyoverlapanddevelop
from pp¯from the decay of ψ(2S,3686) produced by ini-
largetails atlow momenta,mainly due to FSR. Further,
tial state radiation (ISR), and is clearly observed when
theseveralordersofmagnitudelargerQEDyieldsoflep-
tons (e+e− and µ+µ−) overwhelm the hadron peaks. the net vector momentum in the events is increased to
Pp~<150 MeV/c.
The first step of analysis therefore consisted of mini-
mizing the lepton contributions and the radiative tails. The observed cross sections are obtained as σ (s) =
0
This preselection of events was done by removing events N(s)/[ǫ(s) (s)], where ǫ(s) is the MC-determined effi-
withE (energylossinthecentralcalorimeter)/p(track ciency, and·L(s) is the integrated e+e− luminosity. The
CC
momentum)> 0.85 (which reduced e+e− by a factor efficienciesaLt√s=3772MeVand4170MeVarerespec-
105), removingthose withany hits in the muoncham- tively 16.2% and 16.2% for π+π−, 60.2% and 60.9% for
∼bers (which reduced muons by a factor 102), and re- K+K−, and 71.3% and 68.7% for pp¯. These cross sec-
∼
quiring that the vector sum of the momenta of the pair tions are corrected for ISR to obtain the Born cross sec-
be P~p+,− < 60 MeV/c (which greatly reduced the ra- tions using the method of Bonneau and Martin [13]. At
diative tails). To developfurther eventselectioncriteria, √s=3772MeVand4170MeVthecorrectionfactorsare
we use the variable X (E(h+)+E(h−))/√s, as in respectively 0.797 and 0.796 for π+π−, 0.817 and 0.809
h
Ref. [5]. ≡ for K+K−, and 0.806 and 0.800 for pp¯. The Born cross
3
2 2 TABLE I. Cross sections for e+e− π+π−, K+K−, and pp¯
0.0120 s = 3772 MeV (a) PIONS 0.050 s = 4170 MeV (b) PIONS fore+e−annihilationsat√s=3772→MeVand4170MeV,and
s / 100 s / 40 thecorresponding form factors of pion, kaon,and proton.
nt 80 nt
Cou 60 K Cou2300 K pπp¯+π−,K+K− NNπ,pK σσBB ((ppbb)) 1100F2Gπ,MK Q|Q42G|FMπ/,Kµp
40 √s=3772 MeV, Q2 =14.2 GeV2 | |
20 10 π+π− 66|1(2|6) 6.36(25)(36) 0.65(1)(2) 0.92(2)(3)
0 0 K+K− 1564(40) 3.95(10)(22) 0.54(1)(1) 0.76(1)(2)
0.96 0.98 1 1.02 1.04 0.96 0.98 1 1.02 1.04
pp¯ 213(15) 0.46(3)(3) 0.88(3)(2) 0.64(2)(2)
Xπ Xπ
2 2 √s=4170 MeV, Q2 =17.4 GeV2
0.0300 s = 3772 MeV (c) KAONS 0.0120 s = 4170 MeV (d) KAONS π+π− 21|8(1|2) 2.89(16)(16) 0.48(1)(1) 0.84(2)(2)
s / 250 s / 100 K+K− 644(25) 2.23(9)(12) 0.44(1)(1) 0.77(2)(2)
nt200 nt80 pp¯ 92(10) 0.29(3)(2) 0.76(4)(2) 0.82(4)(2)
u u
Co150 π Co60 π
100 40
50 20 and F (17.4 GeV2) = 0.048(1) are based on 661(26)
π
00.96 0.98 1 1.02 1.04 00.96 0.98 1 1.02 1.04 and 218(12) observed counts, respectively. These are
X X listed in Table I. In Fig. 2 we plot our results together
K K
nts / 0.02111024000 s = 3772ψ M(e2VS(e)) PROTONS unts / 0.022205 s = 4170 MeV (f) PROTONS sw|mQuiert2neh|mt=awellniMttphsr2etw(vhJiieot/huψds)i|mQr[e1e2s5n|u]sl.>itosAn,9aisnlGscchloeuoVudwni2nntgianirntgehtreihunielneedfixpigrcrueeecrdlletie,cnrteatisolluanlmgtoreeffoaear--
u80 o15
o C 1/Q2 variation of the form factors at large momentum
C60 10 tra|nsf|ers. Also shown in the figure are three illustrative
40
theoretical predictions. The Q2 behavior of the predic-
5
20
tion of the QCD sum rule inspired model [16] disagrees
0 0
0.96 0.98 1 1.02 1.04 0.96 0.98 1 1.02 1.04 strongly with the data. The pQCD prediction of Gous-
Xp Xp set and Pire [17] is nearly a factor two smaller than our
measurements, and the latest AdS/QCD prediction by
FIG.1. DistributionsofXh [E(h+)+E(h−)]/√sforh π, Brodsky and de Teramond [18] reproduces the data be-
≡ ≡
K, p for √s = 3772 MeV and 4170 MeV data. The vertical low 5 GeV2, but falls to 2/3 of the observed values for
dashed lines bracket the Xh region in which counts are ac- Q2 >5GeV2. Czyzetal.[19]haveshownthatthemea-
cepted. The open histogram in panel (e) corresponds to pp¯ s|ure|d F (Q2 ) at Q2 > 5 GeV2 can be parameterized
π
from ψ(2S) ISR excitation (see text). Arrows indicate ex- | | | |
in a VDM approach,but only by including hypothesized
pected positions of contaminants.
radial ρ , ρ , ρ resonances.
3 4 5
Kaon Form Factors—The results of our present
sectionsarerelatedtothe angleintegratedcrosssections measurements at Q2 = 14.2 GeV2 and 17.4 GeV2 are
| |
listed in Table I. In Fig. 2, we show our present re-
σB(s)π,K =(πα2βπ3,K/3s)|Fπ,K(s)|2, (4) sults for kaon form factors along with early results for
σ (s) =(4πα2/3s)β [G (s)2+(τ/2)G (s)2]. (5) Q2 < 4.4 GeV2 [20], the indirect result for Q2 =
B p p M E | | | |
| | | | M2(J/ψ) [21], and our previous measurement at Q2 =
| |
Thecrosssectionsσ andtheformfactorsF andG 13.48GeV2[5]. Asforpions,allresultsfor Q2 >9GeV2
B π,K M | |
(assuming GE = GM) derived from them are listed in followthepredictedαS/|Q2|behavioroftheformfactors.
Table I. The systematic uncertainties listed in the table No theoretical predictions for kaon timelike form factors
areasdescribedlater. IntheTable,andelsewhereinthis exist. Anempiricalfittothedatahashoweverbeenmade
Letter, the errors indicated in the parentheses are in the by Czyz et al. [19], but it requires an “infinite tower” of
corresponding last digits of the main values. hypothetical ρ, ω, and φ resonances.
Pion Form Factors—Timelike form factors of pions, Proton Form Factors—In Fig. 2, we show the pre-
measured by e+e− π+π−, have been reported by viously published proton form factors for timelike mo-
Babar for 0.09 GeV2→ Q2 8.7 GeV2 by the ISR mentum transfers. At the largest Q2 all these mea-
≤ | | ≤ | |
method[14]. Atthelargest Q2 ,theseresultshaveupto surements were severely limited by statistics, with 0
| |
100% errors, and were based on <10 counts/bin. Our and 1 counts at Q2 = 14.4 GeV2 [4], 16(5) counts at
± | |
earlier measurement [5] at Q2 = 13.48 GeV2 had only Q2 = 13.48 GeV2 [5], and 8(5) counts in the region
26(5)observedcounts,andr|esu|ltedinF (13.48GeV2)= |Q2| = 13.0 14.4 GeV2 [22]. We observe 213(15) and
π
0.075(9). The presentresults, F (14.2 GeV2)=0.065(2) |92(1|0) count−s at Q2 = 14.2 GeV2 and 17.4 GeV2, re-
π
| |
4
3 3 2
BABAR Present π REF. 18 Present K 1.8 BABAR FNAL Present p
222|Q| |F(Q)| (GeV )π12..1255 QCDSR J/ψ AdS/QαCSD 222|Q| |F(Q)| (GeV )K12..1255 J/ψ αS 424µQ||G(Q)|) / (GeV )Mp000111......1468246 Timelike αS2
0.5 0.5 (| Spacelike
pQCD 0.2
0 0 0
0 2.5 5 7.5 10 12.5 15 17.5 20 0 2.5 5 7.5 10 12.5 15 17.5 20 4 6 8 10 12 14 16 18 20
|Q2| (GeV2) |Q2| (GeV2) |Q2| (GeV2)
FIG.2. Summaryofformfactorresults. ThesolidpointsinthepionandprotonpanelsarefromBaBarISRmeasurements[14,
22], the open triangles are from FNAL pp¯measurements [4], the open circles are from the CLEO measurements [5], the open
squares at Q2 =16.1 GeV2 and 18.3 GeV2 are from our small statistics datasets, and thesolid squares are from thepresent
measureme|nts|at Q2 =14.2 GeV2 and 17.4 GeV2. The theoretical curves for pions are from References [16,17,18]. The solid
curvesillustrate th|ea|rbitrarily normalized variation of αS for π and K, and α2S variation for protons.
spectively. This allows us to examine, for the first time, ments for pions, kaons, and protons for the highest
the Q2 dependence of the proton form factors sensi- timelike momentum transfers of Q2 = 14.2 GeV2 and
| | | |
tively. As listed in Table I, and shown in Fig. 2, we find 17.4 GeV2 with nearly five times higher precision than
that in disagreement with the dimensional counting rule priormeasurementsintheliterature. Forpionsandkaons
predictionofthe weakα2 variationof Q4 G (Q2 )/µ , we find that the dimensional counting rule prediction of
S | | M | | p
its value 0.64(3) GeV4 at Q2 = 14.2 GeV2 is 22(4)% α /Q2 variationofthe formfactors with Q2 holdsvery
S
| | | |
smaller than 0.82(5)GeV4 at 17.4 GeV2. We note, how- well. However, the existing theoretical predictions for
ever, that Q2 G (Q2 )/µ is essentially constant, be- pions fail by large factors to predict the magnitude of
M p
ing = 4.5(2|) |10−2| Ge|V2 and 4.7(2) 10−2 GeV2 for the form factors. We find F (Q2 )/F (Q2 ) = 1.21(3)
π K
× × | | | |
Q2 = 14.2 GeV2 and 17.4 GeV2, respectively. This is and1.09(4)for Q2 =14.2GeV2 and17.4GeV2,respec-
| | | |
reminiscent of the fact that for the spacelike momentum tively. TheseareinagreementwithF (Q2 )/F (Q2 )=
π K
| | | |
transfers for the equivalent form factors, F (Dirac) and 1.19(17) measured at 13.48 GeV2 [5], and in large dis-
1
F (Pauli), it is QF /F which is found to be constant, agreement with the asymptotic prediction that they
2 2 1
rather than Q2F /F1 as predicted [7]. should be equal to the ratio of the squares of the decay
2
Recent polarization measurements of the spacelike constants, fπ2/fK2 =0.67(1).
form factors of proton at JLab have revealed that Forprotons,wefindthatthetimelikeformfactorscon-
G (Q2)/G (Q2) monotonically decreases with increas- tinue to be a factor two or more larger than the cor-
E M
ing Q2 [23]. We fit the summed differential cross sec- responding spacelike form factors, as shown in Fig. 2.
tions we measure at √s = 3772 MeV and 4170 MeV We find the unexpected result that Q4 GM(Q2 )/µp at
| | | |
with Eq. 2, and obtain G = (0.85 0.07) 10−2, Q2 = 14.2 GeV2 is 22(4)% smaller than at Q2 =
G = (0.71 1.17) 10−2M, and G /G± = 0×.83+0.98 |17.4| GeV2. The difference is suggestive of the| ne|ar–
atEthe averag±e Q2 ×= 16.1 GeV2. TEhisMshows tha−t0w.6e7 constancy of Q2 GM(Q2 )/µp, instead.
h| |i | | | |
have little sensitivity to G , and even with our larger The overall conclusion of the present investigation is
E
statistics we can not determine G /G . The results that the asymptotic predictions of QCD-based models
E M
listed in Table 1 and shown in Fig. 2 are assuming are not realized even at momentum transfer as large as
G (Q2 ) = G (Q2 ). If G (Q2 ) = 0 is assumed, 18 GeV2, and that theoretical understanding of timelike
E M E
| | | | | |
G (Q2 ) would increase by 6% at Q2 =14.2 GeV2, form factors of hadrons is still lacking at the quantita-
M
| | ∼ | |
and by 5% at Q2 =17.4 GeV2. tive level, and our precision measurements provide new
∼ | |
challenges for theory.
Systematic Uncertainties—As in our previous publi-
This investigation was done using CLEO data, and as
cation[5], weestimate uncertaintiesof1%intriggereffi-
members of the former CLEO Collaborationwe thank it
ciencies,2%intrackingefficiencies,1%inluminosity,and
for this privilege. This research was supported by the
0.2% in radiative corrections. In addition, we estimate
U.S. Department of Energy.
the total uncertainty due to variation of Pp~, E , and
CC
∆ to be <5%for π, K,and p at both √s=3772MeV
L
and 4170 MeV. This brings the total systematic uncer-
tainty to 6%.
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